PHYC10004 Chap.6 Magnetic Forces on Charges and Currents
Magnetic Forces on Charges and Currents
Week 5 introduces the magnetic field through the force it exerts. A charge moving through a field B feels F = qv × B, perpendicular to both its velocity and the field, so the force changes the direction of motion but never the speed.
In a uniform field a charged particle moving across the field travels in a circle of radius mv/qB with a period that does not depend on its speed, the principle behind the cyclotron.
Crossed electric and magnetic fields select a single speed, v = E/B, and a velocity selector feeding a magnetic region becomes a mass spectrometer that separates isotopes by the radius of their paths.
The Hall effect uses the same sideways push to reveal the sign of the charge carriers in a conductor. Summing the force over the moving charges in a wire gives F = iL × B, and on a current loop the forces form a torque τ = μ × B, the working principle of electric motors. Throughout, the direction of the force is a marked part of every answer.
What this chapter covers
- 01
The Magnetic Force Law
- 02
Right-Hand Rules
- 03
Circular Orbits in a Field
- 04
The Cyclotron
- 05
Velocity Selector
- 06
Mass Spectrometer
- 07
The Hall Effect
- 08
Force on a Current
- 09
Torque on a Coil
Worked example · free
Speed and energy of an electron on a measured circle
- 1Magnetic force supplies the centripetal force: evB = mv²/r, so v = eBr/m.
- 1Speed: v = (1.6022 × 10⁻¹⁹)(1.0 × 10⁻³)(0.020)/(9.109 × 10⁻³¹) = 3.5 × 10⁶ m/s.
- 1Kinetic energy: K = ½mv² = 5.6 × 10⁻¹⁸ J, which is 35 eV.
- 1Period: T = 2πm/eB = 3.6 × 10⁻⁸ s, independent of the speed.
Key terms
- Magnetic Field
- The magnetic field B at a point is defined by the force F = qv × B it exerts on a moving charge, measured in teslas.
- Lorentz Force
- The Lorentz force is the total force on a charge in electric and magnetic fields, F = q(E + v × B).
- Cyclotron Period
- The cyclotron period T = 2πm/qB is the time for one circular orbit in a uniform magnetic field, independent of the particle's speed.
- Velocity Selector
- A velocity selector is a region of crossed electric and magnetic fields that lets only particles with speed v = E/B pass undeflected.
- Hall Effect
- The Hall effect is the transverse voltage that builds up across a current-carrying conductor in a magnetic field, whose sign reveals the sign of the charge carriers.
- Magnetic Dipole Moment
- The magnetic dipole moment of a current loop is μ = NiA, directed along the loop's normal by the right-hand grip rule.
Magnetic Forces on Charges and Currents FAQ
Why does a magnetic field not change a charged particle's speed?
The magnetic force is always perpendicular to the velocity, so it does no work on the particle. Its kinetic energy and speed therefore stay constant, while the direction of its velocity, momentum and acceleration keep changing as it curves.
How do I find the direction of the magnetic force on an electron?
Apply the right-hand rule as if the charge were positive: fingers along the velocity, curled towards the field, thumb along v × B. Then reverse the answer, because the electron's negative charge flips the force to the opposite direction.
What path does a charge follow if it enters a field at an angle?
The velocity component along the field is unchanged, while the perpendicular component produces circular motion. The combination is a helix around the field direction, with radius set by the perpendicular speed and pitch equal to the parallel speed times the period.
How does a mass spectrometer separate isotopes?
A velocity selector first passes ions of one speed. In the following uniform field each ion travels a semicircle of radius mv/qB, so at the same speed and charge the radius is proportional to mass and heavier isotopes land farther from the entrance.
Exam move
Make direction practice a daily habit for this chapter: take a set of velocity and field directions, find the force for a proton and then for an electron, and draw all three vectors each time. Then practise the orbit chain from a single force balance, finding radius, speed, period and kinetic energy, including one case with an alpha particle whose charge is 2e.
Work one velocity-selector and spectrometer problem through to the landing separation, and finish with the force on a wire and the torque on a coil, stating which rule gave each direction.
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